Both manuscripts and the 1830 edition have the subordinate conjunction that after reason. The 1837 edition, perhaps unintentionally, omitted the that. There are no other examples of reason postmodified by a finite clause in the Book of Mormon text. However, there is one example of this construction in a revelation given to Joseph Smith in July 1828, during the same general time period as the translation of the Book of Mormon:
Clearly, there is nothing wrong with reason being postmodified by a that-clause. It is quite common in English.
Summary: Restore the subordinate conjunction that after reason in Alma 33:20: “now the reason that they would not look is because they did not believe that it would heal them”.